Hoyt and Fabula 

 For high Reynolds numbers, reasonable approximations are 



s - s* = ks = k'e 

 where k and k' are constants as L is varied. Thus 



A. E. = —-§ X const . 



Co' 



and since 6/9^- 1-R, 



^■^- =c(i%) " <^°"''- 



Thus in boundary-layer applications the additive effectiveness is helped by 

 the reduced boundary- layer throughput as R is increased, but the increase with 

 L is lost. 



Since aconcentrationof a few wppm is 2 to 3 orders of magnitude smaller 

 than used in the pipe-line applications, the newly discovered effectiveness at 

 such concentration of the extremely high molecular weight, linear, soluble poly- 

 mers now makes the situation more hopeful for boundary- layer applications. 

 (Fortunately, for such applications, the extreme sensitivity of the same poly- 

 mers to mechanical degradation may not be a major problem since the use-time 

 of the polymer is short.) However, calculations indicate that even the increase 

 in the factor l/C by about 1000 still leaves the technique of reducing ship fric- 

 tion by boundary- layer additives economically uncompetitive. 



Hence until additive costs can be brought considerably lower, this method 

 of drag reduction appears to be reserved for applications where an emergency 

 speed increase would be required. Of course, in an application where a large 

 proportion of the total drag is frictional, such as a slow speed ship, the tech- 

 nique may look economic. 



In any event, the applications of the rather basic experiments presented 

 here are difficult to foresee. Certainly the possibilities of achieving substantial 

 drag reductions with relatively small amounts of additive are attractive enough 

 to warrant intensive further effort. 



REFERENCES 



1. Toms, B. A., "Some Observations on the Flow of Linear Polymer Solutions 

 Through Straight Tubes at Large Reynolds Numbers," Proceedings of the 

 International Rheological Congress, Scheveningen, Holland, 1948, pp. 

 11-135-41. 



2. Mysels, K. J., "Flow of Thickened Fluids," U.S. Patent 2,492,173, Decem- 

 ber 27, 1949. 



3. Shaver, R. G., and E. W. Merrill, "Turbulent Flow of Pseudoplastic Poly- 

 mer Solutions in Straight Cylindrical Tubes," AM INST CHEM ENGR, J, 

 Vol. 5, No. 2 (1959), p. 181. 



958 



